Bilinear maps on C$^*$-algebras that have product property at a compact element

TL;DR

This paper investigates how bounded bilinear maps with a product property at a specific element influence the structure of C$^*$-algebras, especially compact and von Neumann algebras, and their relation to homomorphism-like maps.

Contribution

It characterizes when C$^*$-algebras are determined by products at a compact element and explores applications to homomorphism-like and derivation-like maps.

Findings

Characterization of C$^*$-algebras determined by products at a compact element

Analysis of bilinear maps with product property in von Neumann algebras with atomic parts

Applications to maps resembling homomorphisms and derivations at fixed points

Abstract

We study bounded bilinear maps on a C$^*$-algebra $A$ having product property at $c\in A$. This leads us to the question of when a C$^*$-algebra is determined by products at $c.$ In the first part of our paper, we investigate this question for compact C$^*$-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.